Homoclinic snaking near the surface instability of a polarisable fluid

Lloyd, David J. B.; Gollwitzer, Christian; Rehberg, Ingo; Richter, R.

doi:10.1017/jfm.2015.565

Cited by 36 publications

(49 citation statements)

References 55 publications

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“…The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence.…”

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confidence: 99%

First order phase transitions and the thermodynamic limit

et al. 2019

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We consider simple mean field continuum models for first order liquid-liquid demixing and solidliquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.However, in a system of finite size or when a finite time horizon is considered, metastable states often play an important role and even unstable states may be crucial, as transient states, for extended time periods. The full set of states and their dependence on the various control parameters is conveniently presented in the form of bifurcation diagrams, well known in the context of dynamical systems and pattern formation theory [3][4][5]. The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence. In the context of buckling the corresponding concept is the Maxwell load [14].In this paper we show and discuss how the discontinuities in the TL represented by the Maxwell construction arise from the bifurcation diagrams relating stable, metastable and unstable steady states in finite-size systems. We focus on two systems: (i) phase decomposition of a binary liquid mixture and (ii) the liquid to crystalline solid phase transition. We investigate the transitions that occur in the context of the most basic mean-field continuum models for these two different phase transitions, namely, the Cahn-Hilliard equation [15][16][17] and the phase field crystal (PFC) model (or conserved Swift-Hohenberg equation) [18][19][20].Some aspects related to this question have been considered previously, in particular in relation to the nature of some of the states that can arise in finite-size systems in the two-phase region. References [21-24] describe theory and computer simulation results for atomistic models exhibiting gas-liquid, liquid-hexatic and hexaticsolid phase transitions that indicate how the Maxwell construction develops as the system size increases or the temperature decreases. For example, figures 3-5 of [23] compare Monte-Carlo computer simulation results in a finite three-dimensional domain (see also figures 2 and 3 of [22]) with the results from a capillary drop type model, also in a finite domain, with a mean-field expression for the chemi...

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First order phase transitions and the thermodynamic limit

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“…Examples include experimental observations in optical systems22, numerical results for plain Couette flow23, as well as experimentally and analytically for hexagonal localised patterns in magnetic fluids24.…”

Section: Diffusion On Complex Networkmentioning

confidence: 99%

Pattern Formation on Networks: from Localised Activity to Turing Patterns

McCullen

Wagenknecht

2016

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Networks of interactions between competing species are used to model many complex systems, such as in genetics, evolutionary biology or sociology and knowledge of the patterns of activity they can exhibit is important for understanding their behaviour. The emergence of patterns on complex networks with reaction-diffusion dynamics is studied here, where node dynamics interact via diffusion via the network edges. Through the application of a generalisation of dynamical systems analysis this work reveals a fundamental connection between small-scale modes of activity on networks and localised pattern formation seen throughout science, such as solitons, breathers and localised buckling. The connection between solutions with a single and small numbers of activated nodes and the fully developed system-scale patterns are investigated computationally using numerical continuation methods. These techniques are also used to help reveal a much larger portion of of the full number of solutions that exist in the system at different parameter values. The importance of network structure is also highlighted, with a key role being played by nodes with a certain so-called optimal degree, on which the interaction between the reaction kinetics and the network structure organise the behaviour of the system.

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“…This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in a good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.Nonlinear temporal pulses and spatial dissipative localized structures appear in various optical, plasma, hydrodynamic, chemical, and biological systems [1][2][3][4][5][6][7][8][9][10][11][12][13]. Being well-separated from each other these structures can interact locally via exponentially decaying tails and, as a result of this interaction, they can form bound states, known also as "dissipative soliton molecules" [14], characterized by fixed distances and phase differences between individual structures.…”

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Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

et al. 2017

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We study the dynamics of an array of nearest-neighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in a good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.Nonlinear temporal pulses and spatial dissipative localized structures appear in various optical, plasma, hydrodynamic, chemical, and biological systems [1][2][3][4][5][6][7][8][9][10][11][12][13]. Being well-separated from each other these structures can interact locally via exponentially decaying tails and, as a result of this interaction, they can form bound states, known also as "dissipative soliton molecules" [14], characterized by fixed distances and phase differences between individual structures. Such bound states can emerge due to the oscillatory character of the interaction force which is related to the presence of oscillating tails. Another scenario occurs in the case of monotonic repulsive interaction when either the pulse tails decay monotonically, or a strong nonlocal repulsive interaction between the pulses is present. In this case the pulses tend to distribute equidistantly in time or space leading to periodic pulse trains [15][16][17][18] which, in contrast to closely packed bound states, exhibit large distances between the consequent pulses.In this Letter we show that even in the case when the pulses in an individual system exhibit strong repulsion, the formation of bound pulse trains can be achieved by arranging several systems in an array with nearestneighbor coupling. As a result, the pulses interact not only within one system, but also with those in the neighboring ones leading to a different balance of attraction and repulsion. More specifically, we demonstrate that this array can produce a periodic train of clusters consisting of two or more closely packed pulses with the possibility to change the interval between the pulses via the variation of coupling phase parameter. We show that the observed pulse train states coexist with the regimes which are amplitude synchronized and possess fixed phase shifts between the pulses emitted by neighboring array elements. In contrast to the pulse bound state regimes predicted and observed experimentally previously [14,[19][20][21][22][23][24][25][26][27][28][29][30][31][32], this regime cannot exist in a solitary pulse-generating system. We illustrate this general result by considering a part...

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Homoclinic snaking near the surface instability of a polarisable fluid

Cited by 36 publications

References 55 publications

First order phase transitions and the thermodynamic limit

First order phase transitions and the thermodynamic limit

Pattern Formation on Networks: from Localised Activity to Turing Patterns

Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

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